Genelleştirilmiş Türevli Yarıasal Halkaların Lie İdealleri

Year 2018, Volume: 8 Issue: 1, 1 - 12, 30.06.2018

Emine Koç Sögütcü , Öznur Gölbaşı

Abstract

R, 2-torsion free bir yarıasal halka ve U, R halkasının bir merkez tarafından kapsanılmayan kare-kapalı Lie ideali olsun. Eğer her x,y∈R için F(xy) = F(x)y + xd(y), koşulunu sağlayan bir d:R→R türevi varsa F dönüşümüne R halkasının d ile belirlenmiş bir genelleştirilmiş türevi denir. Bu çalışmada, aşağıdaki koşullardan biri sağlanırsa d dönüşümünün U üzerinde komüting dönüşüm olduğu gösterilecektir: i) F(u)u = ±uG(u), ii) [F(u),v] = ±[u,G(v)], iii) F(u)∘v = ±u∘G(v), iv) [F(u),v] = ±u∘G(v), v) F([u,v]) = [F(u),v] + [d(v),u]. Burada G:R→R dönüşümü h:R→R türevi ile belirlenmiş bir genelleştirilmiş türevdir.

Keywords

Yarıasal halka, Lie ideal, Türev, Genelleştirilmiş türev

Lie Ideals of Semiprime Rings with Generalized Derivations

Abstract

Let R be a 2- torsion free semiprime ring, U a noncentral square-closed Lie ideal of R. A map F:R→R  is called a generalized derivations if there exists a derivation d:R→R such that F(xy)=F(x)y+xd(y), for all x,y∈R. In the present paper, we shall prove that h is commuting map on U if any one of the following holds: i) F(u)u=±uG(u), ii) [F(u),v]=±[u,G(v)], iii) F(u)∘v=± u∘G(v), iv) [F(u),v]=±u∘G(v), v)F([u,v])=[F(u),v]+[d(v),u] for all u,v∈U, where G:R→R  is a generalized derivation associated with the derivation h:R→R.

Keywords

semiprime ring, Lie ideal, derivation, generalized derivation

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